--- title: "Trigonometry basics" description: "Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem" canonical: https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH source: https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH.md retrieved: 2026-10-19 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Trigonometry basics Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem Subject: Mathematics · Area: Geometry · Ages 13 to 15 Page: https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH ## Ready when they can - Identify which trigonometric ratio to use (SOH CAH TOA) based on the known and unknown sides - Calculate an unknown side using sin, cos, or tan and a known angle - Find an unknown angle using the inverse trigonometric function (e.g. tan⁻¹) ## Lesson: Trigonometry basics: SOH CAH TOA Every right triangle has one 90 degree corner. The side across from it is the hypotenuse, always the longest. Pick one of the two acute angles. The side across from your angle is the opposite. The side touching your angle that is not the hypotenuse is the adjacent. Switch to the other acute angle and opposite and adjacent swap. The hypotenuse never changes. Three ratios connect your angle to its sides. Sin is opposite over hypotenuse. Cos is adjacent over hypotenuse. Tan is opposite over adjacent. SOH CAH TOA keeps them in order. All right triangles with the same acute angle are similar, so these ratios depend only on the angle, not the triangle's size. To find a missing side, pick the ratio naming the side you want and a side you know. If the unknown sits on top, multiply: opposite = hypotenuse times sin(angle). If the unknown sits underneath, divide: hypotenuse = adjacent ÷ cos(angle). To find a missing angle, form the ratio then undo it with the inverse, sin⁻¹, cos⁻¹ or tan⁻¹. The inverse is not a reciprocal. Set your calculator to degree mode first. **Example.** A tree stands vertical on level ground. You stand 20 metres from its foot and sight the top along a straight line from ground level at a 45 degree angle. The ground distance is adjacent, the tree height is opposite, so tan links them. Since tan(45°) = 1, height = 20 × 1 = 20 metres. **Example.** A ramp meets level ground at 60 degrees and its base is 6 metres. The base is adjacent, the ramp length is the unknown hypotenuse, so cos links them: cos(60°) = 6 ÷ length. Since cos(60°) = 0.5, length = 6 ÷ 0.5 = 12 metres. The unknown sat underneath, so we divide. **Example.** A right triangle has an opposite side of 6 m and a hypotenuse of 10 m. To find the angle, form sin θ = 6/10 = 0.6, then undo sin with its inverse: θ = sin⁻¹(0.6) ≈ 36.9°. The inverse is not 1/0.6. Set your calculator to degree mode first. Match your known and wanted sides to one ratio before computing. The unknown's position decides the final step: on top means multiply, underneath means divide. For an angle, build the ratio from two known sides, then apply the matching inverse. Tan is the direct ratio when the two sides in play are opposite and adjacent, but sin or cos can find an unknown hypotenuse by division. **Recap.** Name the sides from your chosen acute angle, pick the ratio naming the wanted and known sides, then multiply or divide by the unknown's position. Use an inverse function in degree mode to find a missing angle. *Adapted from OpenStax Algebra and Trigonometry (CC-BY 4.0), openstax.org Adapted from the licensed first edition. No images imported. The current HTML footer states BY-NC-SA restrictions, but the preserved first-edition adaptation retains its recorded CC BY 4.0 license. Access for free at https://openstax.org/details/books/algebra-and-trigonometry.* · [license](https://creativecommons.org/licenses/by/4.0/) ## Practice 36 questions on this page, each with its working shown. ## Needs first - [Pythagoras' Theorem](https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd) - [Simple formulae](https://lightmysky.com/learn/mathematics/simple-formulae-mt_3qrCtdoVAU) - [Using a Calculator Well](https://lightmysky.com/learn/mathematics/using-a-calculator-well-mt_FZE80D2jS_) - [Similar Triangles and Proving Similarity](https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs) - [Geometric terms and notation](https://lightmysky.com/learn/mathematics/geometric-terms-and-notation-mt_WtIFJSCQIT) ## Opens up - [Forces as Vectors: Resolving and Resultants](https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v) - [Moments and the Equilibrium of a Rigid Body](https://lightmysky.com/learn/mathematics/moments-and-the-equilibrium-of-a-rigid-body-mt_FX5egGL4LI) - [Projectile Motion](https://lightmysky.com/learn/mathematics/projectile-motion-mt_O1C-QxCSJN) - [Resolving Vectors into Components](https://lightmysky.com/learn/science/resolving-vectors-into-components-mt_tc698unGJG)